314 ANALYTICAL Ml* 'HANK'S a sphere of 4000 miles radius and the Krnvit;ilinn;d force to vary inversely as the square of the distance from the center of the earth. 6. Given the height of a mountain above the surrounding plain and the period of a pendulum on the plain and on the fop of (he mountain, find a relation from which the radius of the earth enu he computed. 6. Supposing the gravitational attraction within the earth to vary aa the distance from the center, find the depth below the surface at which a seconds pendulum will beat 2 seconds. 7. Derive a relation between the distance of a pendulum from the center of the earth and its period. 8. A balloon ascends with a constant acceleration and reaches 400 feet in one minute. What is the rate at which the pendulum Rains in the bal- loon? 9. A pendulum of length / is shortened by a small amount o/. Show that it will gain about nj.: vibrations in an interval of time of n vibra- tions. n is supposed to be a large integral number. 10. How high above the surface of the1 earth must a seconds pendulum be carried in order that it may have a period of I seconds? 11. While a train is taking a curve at the rate of <io mil«\s per hour a seconds pendulum hanging in the train Is observed in .swing at the rale of 121 oscillations in 2 minutes. Show that the radius of the curve is about a quarter of a mile. 12. Find the expressions for the lens! period of oscillation the following bodies can have; also determine the corresponding portion of the axes. (a) Rod of negligible transverse dimensions. (d) Solid cylinder. (b) Square plate of negligible thickness. (e) Solid .sphere. (c) Circular plate of negligible thickness. If) Spherical shell. 244. Determination of the Gravitational Acceleration by Means of a Reversible Pendulum. A physical pendulum which is provided with two convenient axes of vibration is called a reversible pendulum. Lot />un«I />', MR*. 1 41, denote the distances of the axes from the center of muss. Then the corresponding periods are